TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

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Bibliography [Bou71] , Eléments de Mathématiques. Groupes et Algèbres de Lie, Vol. Chapitre I, Algèbres de Lie, Hermann, Paris, 1971. ↑133, 134 [BP89] H. Benamor and G. Pinczon, The graded Lie algebra structure of Lie superalgebra deformation theory, Lett. Math. Phys. 18 (1989), no. 4, 307 – 313. ↑62, 64, 66, 74 [Bur06] D. Burde, Classical r-matrices and Novikov algebras, Geom. Dedicata 122 (2006), 145–157. ↑116 [CM93] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reihnhold Mathematics Series, New York, 1993. ↑viii, 3, 5, 8, 10, 12 [Dix74] J. Dixmier, Algèbres enveloppantes, Cahiers scientifiques, fasc.37, Gauthier-Villars, Paris, 1974. ↑35 [DPU] M. T. Duong, G. Pinczon, and R. Ushirobira, A new invariant of quadratic Lie algebras, Alg. Rep. Theory (to appear). ↑ [FK94] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford Mathematical Monographs, 1994. ↑99 [FS87] G. Favre and L. J. Santharoubane, Symmetric, invariant, non-degenerate bilinear form on a Lie algebra, J. Algebra 105 (1987), 451–464. ↑viii, 17, 19, 20, 39 [GD79] I. M. Gel’fand and I. Y. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl 13 (1979), no. 4, 248 – 262. ↑xvii [Gié04] P. A. Gié, Nouvelles structures de Nambu et super-théorème d’Amitsur-Levizki, Thèse de l’Université de Bourgogne (2004), 153 pages. ↑61, 64, 66 [Hum72] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Maths., Springer-Verlag, New York, 1972. ↑6 [Hum95] , Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, 1995. ↑3 [Jac51] N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc 70 (1951), 509–530. ↑ [Kac85] V. Kac, Infinite-dimensional Lie algebras, Cambrigde University Press, New York, 1985. ↑vii, 17, 19 [Kos50] J-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bulletin de la S. M. F 78 (1950), 65–127. ↑ [Mag10] L. Magnin, Determination of 7-dimensional indecomposable Lie algebras by adjoining a derivation to 6-dimensional Lie algebras, Alg. Rep. Theory 13 (2010), 723 – 753. ↑36 [MPU09] I. A. Musson, G. Pinczon, and R. Ushorobira, Hochschild Cohomology and Deformations of Clifford- Weyl Algebras, SIGMA 5 (2009), 27 pp. ↑xii, 61, 62 [MR85] A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant, Ann. Sci. École Norm. Sup. 4 (1985), 553–561. ↑vii, 17, 19 [NR66] A. Nijenhuis and R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1–29. ↑63 [Oom09] A. Ooms, Computing invariants and semi-invariants by means of Frobenius Lie algebras, J. of Algebra 4 (2009), 1293 – 1312. ↑36 [Ova07] G. Ovando, Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skewsymmetric maps, J. Algebra and its Appl. 6 (2007), no. 6, 897–917. ↑xi, xii, 59, 108, 119 [PU07] G. Pinczon and R. Ushirobira, New Applications of Graded Lie Algebras to Lie Algebras, Generalized Lie Algebras, and Cohomology, J. Lie Theory 17 (2007), no. 3, 633 – 668. ↑vi, vii, 17, 18, 22, 27, 28, 35, 36, 59, 66 [Sam80] H. Samelson, Notes on Lie algebras, Universitext, Springer-Verlag, 1980. ↑132 142
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Université de Bourgogne - Dijon UF
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Remerciements Je remercie l’Ambas
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Contents 3.4.2 Quadratic dimension
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Introduction structures can be dete
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Introduction and Di red (n + 2) th
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Introduction and obtain the followi
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y Introduction We realize that with
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Introduction by I. Bajo, S. Benayad
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Introduction (2) there exists an i-
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xix
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Notations 0 Ik where J = ∈ gl(2
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1.1. Definitions and gε(V ) = {C
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1.2. Nilpotent orbits Lie algebra g
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1.2. Nilpotent orbits where µi = i
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1.2. Nilpotent orbits The point of
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1.3 Semisimple orbits We recall the
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1.4. Invertible orbits Let us now c
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1.5. Adjoint orbits in the general
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2.1. Preliminaries (3) Set the map
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2.1. Preliminaries Proof. Let I be
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2.2. Singular quadratic Lie algebra
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If l = k then we stand at (k − 1)
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Consider Then dim(g) = 6 ∑ i=3 2.
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(2) g g ′ if and only if g i g
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2.3. Quadratic dimension of quadrat
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2.4. 2-step nilpotent quadratic Lie
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Chapter 3 Singular quadratic Lie su
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3.1. Application of Z × Z2-graded
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3.2. The dup-number of a quadratic
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3.3. Elementary quadratic Lie super
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3.3. Elementary quadratic Lie super
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3.4. Quadratic Lie superalgebras wi
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3.5. Singular quadratic Lie superal
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3.5. Singular quadratic Lie superal
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Page 117 and 118: Chapter 4 Pseudo-Euclidean Jordan a
Page 119 and 120: 4.1. Preliminaries Proof. The asser
Page 121 and 122: 4.1. Preliminaries T (x,y) = φ(x)(
Page 123 and 124: 4.1. Preliminaries (3) π(xx ′ )y
Page 125 and 126: 4.2. Jordanian double extension of
Page 127 and 128: 4.2. Jordanian double extension of
Page 129 and 130: 4.3. Pseudo-Euclidean 2-step nilpot
Page 133 and 134: Proof. 4.3. Pseudo-Euclidean 2-step
Page 137 and 138: 4.4. Symmetric Novikov algebras (2)
Page 139 and 140: 4.4. Symmetric Novikov algebras Lem
Page 145 and 146: 4.4. Symmetric Novikov algebras Lem
Page 147 and 148: 4.4. Symmetric Novikov algebras Pro
Page 149 and 150: Appendix A In this appendix, we rec
Page 151 and 152: Appendix A Lemma A.4. Assume that C
Page 153 and 154: Appendix B Here we prove: Lemma B.1
Page 155 and 156: Appendix C We will classify (up to
Page 157 and 158: Appendix C Lemma C.3. The element I
Page 159 and 160: Appendix D In the last appendix, we
Page 161: Bibliography [AB10] I. Ayadi and S.
Page 165 and 166: Index Bε, 3 E(g,B,δ), 53 Iε(V ),
Page 167 and 168: normal form, 5 Jordan-admissible, 1
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